Magnetic Domains and Surface Effects in Hollow Maghemite Nanoparticles

In the present work, we investigate the magnetic properties of ferrimagnetic and noninteracting maghemite (g-Fe2O3) hollow nanoparticles obtained by the Kirkendall effect. From the experimental characterization of their magnetic behavior, we find that polycrystalline hollow maghemite nanoparticles are characterized by low superparamagnetic-to-ferromagnetic transition temperatures, small magnetic moments, significant coercivities and irreversibility fields, and no magnetic saturation on external magnetic fields up to 5 T. These results are interpreted in terms of the microstructural parameters characterizing the maghemite shells by means of an atomistic Monte Carlo simulation of an individual spherical shell model. The model comprises strongly interacting crystallographic domains arranged in a spherical shell with random orientations and anisotropy axis. The Monte Carlo simulation allows discernment between the influence of the structure polycrystalline and its hollow geometry, while revealing the magnetic domain arrangement in the different temperature regimes.


I. INTRODUCTION
In extended materials, the strength and length scale of typical spin-spin interactions are such that ordering of spins frequently occurs over ranges with sizes in the nanometer scale. In nanoparticles, however, the crystal size and geometry determine the extent and configuration of the magnetic domains. In polycrystalline nanostructures and nanoparticle arrays, the competition between the crystallographic anisotropy and the strength of the spin-spin interaction between neighboring crystals, determines the magnetic behavior of the composites. This competition relies not only on the size and shape of the crystallographic domains, but also on their relative orientation and geometric organization.
Due to such dependencies of the magnetic properties, advances in the ability to pattern matter on the nanometer scale have created new opportunities to develop magnetic materials with novel characteristics and applications. [1][2][3][4] One such novel type of magnetic material design, which has recently attracted significant attention, [5][6][7][8] is the hollow geometry. D. Goll et al. showed that the hollow geometry incorporates additional parameters for the tuning of the magnetic properties of the nanoparticles. 6 They theoretically determined the phase diagram of the lowest-energy domain configurations in hollow ferromagnetic nanoparticles as a function of the material parameters, particle size and the shell thickness. 6 However, while this initial model did not include interface or surface effects, actual hollow nanoparticles are characterized by large surface to bulk ratios. Moreover, hollow nanoparticles synthesized by the Kirkendall effect [8][9][10][11] or by means of templates, [12][13][14] are usually polycrystalline structures, due to the multiplicity of shell nucleation sites. Thus, they have multiple crystallographic domains, which are randomly oriented and so have differentiated local anisotropy axes.
In the present work, we study the magnetic properties of polycrystalline hollow maghemite nanoparticles obtained by the Kirkendall effect. We experimentally analyze their magnetic behavior and interpret our experimental results using an atomistic Monte Carlo simulation of a model for an individual maghemite nanoshell.

II. HOLLOW NANOPARTICLES AND STRUCTURAL CHARACTERIZATION
Hollow maghemite nanoparticles were obtained following a previously reported procedure based on the Kirkendall effect. 8 Briefly, iron pentacarbonyl was decomposed in air-free conditions at around 220ºC in organic solvents containing surfactants. The resulting iron-based nanoparticles were oxidized in solution by means of a dry synthetic air flow. Owing to the faster self-diffusion of iron than oxygen ions within iron oxide, the oxidation of 1-20 nm iron nanoparticles results in hollow iron oxide nanostructures.
Hollow iron oxide nanoparticles obtained by the Kirkendall effect have an inner-to-outer diameter ratio of around  I / E = 0.6 and relatively narrow particle size distributions. The hollow nanoparticles studied in this work have a diameter of 8.1±0.6 nm with 1.6±0.2 nm thick shells and a size dispersion of around 10%. Figure 1(a) shows a transmission electron micrograph of the hollow iron oxide nanoparticles supported on a carbon grid.
Further high resolution TEM characterization of the particles show them to be crystalline, but to contain multiple crystallographic domains within each shell ( Fig. 1(b) and 1(c)).
Each hollow nanoparticle is composed of approximately 10 crystallographic domains having random orientations. The presence of intergrains in the shell may allow for surfactants and solvent to enter inside the particle, thus there may not be a true void inside these structures, but it may be filled with organic solvents in solution and with gas in air. The crystallographic structure of the hollow nanoparticles was identified as that of maghemite by X-ray absorption spectroscopy. 8

III. MAGNETIC PROPERTIES
Prior to magnetic characterization, the solution containing the maghemite shells was centrifuged to remove any possible particle aggregates. For magnetic characterization, maghemite nanoshells were dispersed in a 50% mixture of high melting point organic solvents, namely: nonadecane (C 19 H 40 , T m = 32 ºC) and dotriacontane (C 32 H 66 , T m = 69 ºC). In order to avoid interparticle interactions, the particle concentration was kept at about 0.2-0.3 % in mass, as measured by means of ionic-coupled mass spectroscopy. The magnetic measurements were carried out in an XL Quantum Design superconducting quantum interference device (SQUID) using 0.2 g of the diluted sample. Figure 2 shows the magnetic susceptibility vs. temperature for the maghemite nanoshells following zero-field-cool (ZFC) and field-cool (FC) processes. The close coincidence of the ZFC peak and the onset of the irreversibility between the ZFC and FC magnetization curves allow us to exclude a large extent of particle aggregation or large size distributions, which is consistent with the TEM characterization of the sample (see Fig.   1). For the low concentration range used in our experiments, the temperature at the ZFC peak is about 34 K and independent of the particle concentration, which excludes interparticle interactions. 15 This value of the temperature of the ZFC peak is lower than the blocking temperature observed in 7 nm solid maghemite particles, which have a particle volume, and thus a number of spins, equivalent to that of the 8.1 nm hollow particles (roughly 200 nm 3 and 8x10 3 Fe atoms per particle). 16 However, this value of the temperature of the ZFC peak is larger than that corresponding to isolated maghemite crystallites of about 21 nm 3 (about 3.4 nm in diameter assuming spherical shape), equivalent in size to those forming the shell (inset to Figure 2). This experimental observation indicates that either (a) magnetic interactions among crystallites within each hollow particle yield magnetic frustration, which increases the effective blocking temperature of the crystallite, or (b) there is an enhanced value of the anisotropy energy per unit volume with respect to that of solid nanoparticles with similar magnetic volumes. The study of the particle magnetization as a function of the observational time window, by means of ac susceptibility measurements, is a conventional method to evaluate the average magnetic anisotropy barrier per particle (Fig. 3). For a given measuring frequency () and particle size distribution, the real part of the ac susceptibility (') peaks at a temperature (T max ) such, that the measuring time (=1/) coincides with the relaxation time of those magnetic domains having the average anisotropy energy and size. Taking into account that T max and the attempt time are related through the Arrhenius' law, the mean value of the anisotropy energy can be evaluated by linear regression of  as a function of 1/T max (see inset to Fig. 3). For the hollow particles, this regression yields an anisotropy energy per unit volume of 7x10 6 erg/cm 3 . Such a magnetic anisotropy constant is one order of magnitude larger than that of solid nanoparticles with a similar number of spins (7 nm in diameter assuming spherical shape), 17 and two orders of magnitude larger than that of bulk maghemite (4.7x10 4 erg/cm 3 ). 18 It is commonly agreed that, at the surface, the broken translational symmetry of the crystal and the lower coordination leads to a stronger anisotropy than in the bulk.
Anisotropy energies per atom at the surface are usually two or three orders of magnitude larger than in bulk materials, yielding an anisotropy enhancement in nanoparticles and thin films. [19][20][21] . Thus, we associate the huge particle anisotropy obtained for hollow maghemite nanoparticles to the large proportion of spins with lower coordination, located at the innermost or outermost surfaces of the shell and at the interfaces between crystallographic domains. for single crystal nanoshells of diameter below 10-20 nm. 6 Besides, there also exists a significant high-field linear contribution to the magnetization, arising from the spins at the shell surface and crystallite interfaces, which are strongly pinned along local axes due to surface anisotropy. The saturation magnetization associated with the spins at the crystallite cores, which are those remaining with ferrimagnetic ordering like in bulk maghemite, can be estimated to be about 3-4 emu/g by linear extrapolation to zero field of the hysteresis loop at high fields ( Fig. 4(a)). This value is about 20 times smaller than that corresponding to the bulk counterpart (74 emu/g), what gives a clear indication of the high magnetic frustration and high fraction of surface spins present in the hollow particles. Such magnetic frustration, arising from the existence of magnetic domains and surface anisotropy effects, is at the origin of the observed high irreversibility and coercive field of the polycrystalline hollow nanoparticles. In addition, a strong shift of the hysteresis loop, over 3000 Oe, is observed when cooling the particles in the presence of a magnetic field. Note that, in these experiments, the maximum applied field is lower than the irreversibility field, so the observed loop shift may not correspond to an exchange bias phenomenon, but just to a minor loop of the hysteresis loop.
where m is the magnetic moment per particle and  p is a paramagnetic susceptibility ( Fig.   4(b)). The obtained distribution of magnetic moments P(m) is shown in the inset to Fig.   4(b). The mean magnetic moment per hollow particle of this distribution is 3.3x10 -18 emu (360  B , where  B is the Bohr magneton). This magnetic moment is equivalent to 9 nm 3 of bulk maghemite (74 emu/g), which is a volume 24 times smaller than that of the total material volume per hollow nanoparticle. It is worth noting that the saturation magnetization of the ferrimagnetic component deduced from this fitting is about 3 emu/g, which is in good agreement with the value estimated from the hysteresis loop. From the fitting of the magnetization curve at 200 K, a large paramagnetic susceptibility  p is also obtained (see linear contribution in Fig. 4(b)). This very large high-field susceptibility is consistent with the shape of the hysteresis loop at 5 K.
The very low saturation magnetization and the high paramagnetic susceptibility are explained by the large disorder on the hollow nanoparticles ubiquitous surface and crystallographic interfaces, which leads to the reduction of the number of spins aligning with the external field. 22,23 Furthermore, aside from the spin disorder at the nanoparticles surface, the ferrimagnetic character of maghemite has associated significant finite size effects: [24][25][26] Maghemite's net magnetic moment arises from the unbalanced number of spins in an antiparallel arrangement. In the nanoscale, this balance can differ from that of the bulk material, leading to a significant reduction of the magnetization.
From an experimental point of view, the shell magnetization can be increased by improving the shell crystalline structure in two ways: i) An increase of the synthesis temperatures or a-posteriori sintering process would lead to less defective and larger crystallographic domains. However, the growth of the crystallographic domains within the shell is limited by the shell thickness and thus by the particle size. An excessive growth of the crystallographic domains within the shell leads to its rupture. 8 The first term is the nn exchange interaction, the second is the Zeeman energy with h= μH/k B (H is the magnetic field and μ the magnetic moment of the magnetic ion), and the third corresponds to the magnetocrystalline anisotropy energy. 27 In this last term, we have distinguished surface spins, having reduced coordination with respect to bulk and anisotropy constant k S , from the core spins, having full coordination and an anisotropy constant k C . We consider a Neél type anisotropy for the surface spins and a uniaxial anisotropy along the direction $ i n for the core spins. The corresponding energy can be expressed as: where ij r $ is a unit vector joining spin i with its nearest neighbors j and $ i n is the anisotropy axis of each crystallite. The simulated hollow spherical nanoparticles have a total radius of 4.88 a (where a is the cell parameter of the maghemite) and a shell with thickness D Sh varying between 1.92 a (actual thickness of the hollow particles experimentally studied in this work) and 4.88 a (filled particle). In order to better model the structure of the real particles, the spherical nanoshell has been divided into 10 crystallites having approximately the same volume and number of spins, as depicted in the scheme of Fig. 5.
Every crystallite has a different uniaxial anisotropy direction $ i n taken at random. As for the values of the anisotropy constants, we have taken K C = 4.710 4 erg/cm 3 (the value corresponding to bulk maghemite) and have evaluated K S = 0.1-1 erg/cm 2 by considering the effective anisotropy obtained from the magnetization measurements as eff C S S K = K + K V (being V and S the particle volume and surface, respectively). When expressed in units of K/spin, as used in the simulations, these values translate to k C ≥ 0.01 K and k C ≥1-5 K. Note that hollow polycrystalline particles, like the ones experimentally analyzed here, have 8950 spins, from which 91% are surface spins.
In Fig. 5, we display a snapshot of the low temperature magnetic configuration for k S = 30 K attained after cooling from a disordered high temperature phase in zero applied magnetic field. The spins corresponding to each crystallite are colored differently and, inside each crystal, core spins have been distinguished with a lighter color tone. 28 Inspection of the displayed configuration shows that core spins tend to order In order to demonstrate the peculiar magnetic behavior of the nanoparticles associated to their hollow structure, we have simulated hysteresis loops for polycrystalline particles with different shell thicknesses; from a solid particle to a hollow particle with shell thickness similar to those of the particles experimentally characterized in this work. The hysteresis loops at low temperature (T= 0.5 K), were simulated by cycling the magnetic field between h= 100 K in steps of 1 K. In figure 6, such hysteresis loops are shown for The role of an increased surface anisotropy with respect to bulk for a hollow particle with the real dimensions can be understood by looking at the hysteresis loops computed for different values of k S shown in Fig. 7. When increasing surface anisotropy, the loops become more elongated, and they have lower high field susceptibility and higher closure fields. The qualitative shape of the loops for k S > 10 K becomes similar to that of the measured ones shown in Fig. 4(a), demonstrating that the magnetization dynamics of real samples is dominated by the high proportion of spins on the outer regions of the crystallites forming the shell and their increased surface anisotropy. Moreover, by looking at the contribution of the core spins presented in panel (b) of Fig. 7, we see that the hysteresis loop of the core spins changes from square shaped to elongated with increasing k S , indicating the increasing influence of the disordered surface spins on the reversal mode of the individual crystallites and of the whole hollow particle, which confirms the previous conclusion.
In Fig. 8, the simulated hysteresis loops obtained after field cooling the particle from a high temperature disordered state down to T= 0.5 K in different fields are shown. From these simulations, an appreciable shift of the hysteresis loop towards the left of the applied field axis can be observed for k S = 30 K. Similar shifts were also experimentally obtained after field cooling the hollow particles. This loops shift is certainly due to the fact that, for high k S values, the applied field is not enough to saturate even the core spins. Therefore, the computed loop is a minor loop and the shift should not be erroneously ascribed to any exchange bias effects.    function of the average volume of material per particle (for hollow particles, the average volume of the cavity has been substracted from the average total particle volume). In this analysis, the point corresponding to the peak of the dc ZFC curve has also been included assuming a characteristic time window for that experiment of about 50 s. This point is distinctively marked as an empty circle.      The magnetic field is measured here in temperature units, h= μH/k B , where μ is the atomic magnetic moment.

Conclusions
28 See http://www.ffn.ub.es/oscar/Hollows/Hollows.html for a higher resolution version of this figure.